Solution methods for the p-median problem: An annotated bibliography

Networks

Volumn 48, Issue 3, 2006, Pages 125-142

  Reese, J ^a  

^a UNIVERSITY OF HOUSTON   (United States)

Author keywords

Facility location; p median problem

Indexed keywords

FACILITY LOCATION; P-MEDIAN PROBLEMS;

BIBLIOGRAPHIES; INTERCONNECTION NETWORKS;

PROBLEM SOLVING;

EID: 33845774430     PISSN: 00283045     EISSN: 10970037     Source Type: Journal    
DOI: 10.1002/net.20128     Document Type: Article

References (132)

1. O. Alp, E. Erkut, and Z. Drezner, An efficient genetic algorithm for the p-median problem, Ann Oper Res 122 (2003), 21-42. Describes a simple and fast genetic algorithm that models the indices of vertices in the solution as genes of a chromosome. The fitness function is the objective function. Whereas traditional genetic methods use a crossover approach, this method creates a union of the parents' chromosomes, creating an infeasible solution with m > p genes. A greedy deletion heuristic is applied to decrease the number of genes until p genes are left. No mutation operator is used.
2. A. Archer, R. Rajagopalan, and D.B. Shmoys, Lagrangian relaxation for the k-median problem: New insights and continuity properties, Proc 11th Ann Eur Symp Algorithms, Springer-Verlag, Berlin, 2003, pp. 31-42. Shows that a modification to the algorithm in [71] attains a "continuity" property. The paper also shows that an algorithm originally developed for the UFLP can be used for the p-median problem due to its capability of preserving the Lagrangian multipliers. This algorithm is described and a proof of its approximation factor of 3 is presented.
3. A. Ardalan, A comparison of heuristic methods for service facility locations, Int J Oper Production Manage 8 (1988), 52-58. Introduces a heuristic for a service facility location problem. In [101], this heuristic is adapted to (hepmedian problem and compared to vertex substitution [126].
4. V. Arya, N. Garg, R. Khandekar, V. Pandit, A. Meyerson, and K. Munagala, Local search heuristics for k-median and facility location problems, SIAM J Comput 33 (2004), 544-562. Defines locality gap as the maximum ratio of a locally optimal solution to the global optimum. The fact that local search, constrained by swapping only one vertex at a time, has a locality of gap of 5 is established. When the search is allowed to swap up to p vertices at a time, the locality gap improves to 3 + 2/p.
5. J. Ashayeri, R. Heuts, and B. Tammel, A modified simple heuristic for the p-median problem, with facilities design applications, Robotics Comput-Integrated Manufact 21(2005), 451-464. A simple modification to vertex substitution [126] is presented. The authors show that in some cases, this modification leads to improved solutions. The modified method, called adjusted vertex substitution (AVS), is implemented with multiple starting points to avoid finding local optima. Computational results show that these methods are generally able to improve the solutions found with the original vertex substitution method.
6. P. Avella and A. Sassano, On the p-median polytope, Math Program 89 (2001), 395-411. Defines two classes of facet-defining inequalities and uses them in a cutting-plane algorithm. The authors report using this method to solve several test problems to optimality.
7. P. Avella, A. Sassano, and I. Vasil'ev, Computational study of large-scale p-median problems, Technical report 08-03, Università di Roma "La Sapienza," 2003. Presents a branch-and-price-and-cut algorithm to solve large-scale instances of the p-median problem. This involves a column-and-row generation strategy to solve a relaxed LP and cutting planes to strengthen the formulation. The authors report computational results on large problems involving at least 900 vertices.
8. P. Avella, A. Sassano, and I. Vasil'ev, A heuristic for large-scale p-median instances, Electron Notes Discrete Math 13 (2003), 14-17. Presents a three-step heuristic for large p-median problems. First, a Lagrangian relaxation is solved by using a subgradient algorithm, producing a lower bound. Second, a subset of "promising" variables is selected to create a core problem. Third, branch-and-bound is run on the core problem to find an upper bound. Computational results comparing this method with the GRASP metaheuristic [104] are presented.
9. P. Avella and A. Sforza, Logical reduction tests for the p-median problem, Ann Oper Res 86 (1999), 105-115. Proposes reduction tests for the p-median problem and shows their impact on problem size. These tests are based on characteristics of p-median solutions. These tests are implemented in Lagrangian heuristics and used to solve randomly generated instances and problems from the OR-Library.
10. K.R. Baker, A heuristic approach to locating a fixed number of facilities, Logist Transportation Review 10 (1974), 195-205. Presents a heuristic based on dynamic programming concepts for the p-median problem with fixed p. This is a suboptimal method concerned only with obtaining a good solution rapidly.
11. M. Balinski, Integer programming, methods, uses and computation, Manage Sci 12 (1965), 253-313. Provides an early integer programming formulation of the plant location problem that has historically been adapted to the p-median problem.
12. J.E. Beasley, A note on solving large p-median problems, Eur J Oper Res 21 (1985), 270-273. Enhances the tree search algorithm in [27]. The enhanced algorithm was combined with a Cray supercomputer to solve large problem instances (up to 900 vertices).
13. J.E. Beasley, Lagrangean heuristics for location problems, Eur J Oper Res 65 (1993), 383-399. Introduces a Lagrangian heuristic that combines vertex substitution [126] with subgradient optimization and the methods in [30] and [97]. Computational results are presented comparing this method to standalone vertex substitution.
14. C. Beltran, C. Tadonki, and J.P. Vial, Solving thep-median problem with a semi-Lagrangian relaxation, Technical report, Logilab, HEC, University of Geneva, 2004. Describes a semi-Lagrangian relaxation that, under certain conditions, closes the integrality gap for any linear combinatorial problem with equality constraints. For large enough Lagrangian multipliers, this relaxation closes the integrality gap. The p-median problem is easier to solve, however, when the multipliers are small. The insight here is to increase the Lagrangian multipliers but keep them small enough to keep an associated subproblem easy until an integer optimal solution is found. Computational results are presented, showing that this method solved to optimality several "easy" problems, and improved the best known dual bounds on several unsolved "difficult" problems. The method was able to solve to optimality one of the previously unsolved "difficult" problems.
15. 2).
16. B. Bozkaya, J. Zhang, and E. Erkut, "An efficient genetic algorithm for the p-median problem," Facility location: Applications and theory, Z. Drezner and H. Harnacher (Editors), Springer Verlag, Berlin, 2002, pp. 179-205. A genetic algorithm that models solutions with chromosomes is developed. Each gene of the chromosome is an index of a p-median vertex. Three crossover operators are presented and tested. These variations are compared with the genetic algorithm in [68] and favorable results are reported. The genetic algorithm here improves solutions slowly but steadily, and the authors claim that it is less likely than vertex substitution [126] to be trapped in local optima.
17. M.E. Captivo, Fast primal and dual heuristics for the p-median location problem, Eur J Oper Res 52 (1991), 65-74. Combines the greedy algorithms of [30] and [78] with the alternate heuristic in [87], creating a new heuristic called GreedyG. The author develops a dual-based procedure based upon the method in [47] and a heuristic that finds a primal solution based upon a good dual solution and complementary slackness conditions. These techniques are used to develop bounds to compare the results of several heuristics. GreedyG is compared to the alternate heuristic [87], vertex substitution [126], and hybrid techniques formed by merging a greedy heuristic with the alternate and vertex substitution heuristics.
18. A. Ceselli, Two exact algorithms for the capacitated p-median problem, 40R: Q J Belgian, French Italian Oper Res Soc 1 (2003), 319-340. Introduces branch-and-bound and branch-and-price methods for solving the capacitated variation of the /j-median problem. The branch-and-bound technique uses Lagrangian relaxation and subgradient optimization, and the branch-and-price algorithm uses column generation. The authors note that the ratio p/n strongly affects the behavior of these algorithms. The performance of the branch-and-bound technique worsened quickly as the ratio increased. The branch-and-price algorithm was found to be more stable.
19. A. Ceselli and G. Righini, A branch-and-price algorithm for the capacitated p-median problem, Networks 45 (2005), 125-142. Presents two IP formulations of the capacitated p-median problem and compares the lower bounds on the two corresponding LP relaxations. The authors offer two separate branching strategies for a branch-and-price algorithm. The first is branching on binary variables, and the second is branching on semiassignment constraints. The latter involves partitioning the set of candidate medians for each vertex into two subsets of balanced cardinality and branching in these subsets. The authors found this method to be more effective than branching on binary variables. The authors also present several pricing methods and extensive experimental results.
20. M. Charikar, C. Chekuri, A. Goel, and S. Guha, Rounding via trees: Deterministic approximation algorithms for group Steiner trees and k-median, STOC '98: Proc Thirtieth Ann ACM Symp Theory Comput, ACM Press, New York, 1998, pp. 114-123. A deterministic approximation algorithm that matches the best known randomized approximation is presented. This algorithm has an approximation ratio of O(log p log log p).
21. ij, or the metric distance.
22. M. Charikar, S. Guha, E. Tardos, and D.B. Shmoys, A constant-factor approximation algorithm for the k-median problem (extended abstract), STOC '99: Proc Thirty-First Ann ACM Symp Theory Comput, ACM Press, New York, 1999, pp. 1-10. Provides a 20/3-approximation algorithm for the metric p-median problem. The authors claim that this is the first constant factor approximation algorithm presented for thep-median problem. The algorithm solves a relaxed LP and builds a collection of trees from the solution. It then solves the problem optimally on this collection.
23. th harmonic number.
24. Y. Chiou and L.W. Lan, Genetic clustering algorithms, Eur J Oper Res 135 (2001), 413-427. Applies cluster seed points to solve the p-median problem. A genetic algorithm is used to select the most suitable cluster seeds (medians). The remaining vertices are assigned to clusters according to their similarity to the cluster seeds or their ability to improve the objective function. Computational results are presented.
25. F. Chiyoshi and R.D. Galvão, A statistical analysis of simulated annealing applied to thep-median problem, Ann Oper Res 96 (2000), 61-74. Presents an algorithm that combines vertex substitution [126] with simulated annealing. The algorithm uses vertex substitution to find pairs of vertices to consider for possible exchange, instead of randomly choosing pairs of vertices. The authors adopt a cooling structure that allows for temperature adjustments rather than just temperature reductions.
26. i at any stage. A method for calculating a lower bound on the objective function to further limit the search is also presented.
27. N. Christofides and J.E. Beasley, A tree search algorithm for the p-median problem, Eur J Oper Res 10 (1982), 196-204. Uses the Lagrangian duals of two LP relaxations and subgradient optimization to develop two lower bounds. These bounds are combined with upper bounds found with a heuristic to develop penalty tests that often reduce the size of the problem. The bounds and penalty tests are incorporated within a tree search algorithm. Computational results are presented, comparing this method to the one in [47].
28. M. Chrobak, C. Kenyon, and N. Young, The reverse greedy algorithm for the metric k-median problem, Informat Process Lett 97 (2006), 68-72. Describes a reverse greedy algorithm that is similar to the drop algorithm in [129]. The RGreedy algorithm begins with all n vertices in the solution set and at each step removes the vertex that minimizes the increase in the objective function. The algorithm terminates when k vertices remain in the solution set. The authors show that if the problem is defined on a metric space this algorithm has an approximation ratio between Ω (log n/log log n) and O(log n).
29. R.L. Church, COBRA: A new formulation of the classic p-median location problem, Ann Oper Res 122 (2003), 103-120. Presents an alternate p-median formulation called COndensed Balinski constraints with the Reduction of Assignment variables (COBRA). As the name suggests, this formulation is usually smaller than the classic p-median formulation of [11], reducing the number of variables by up to 60%. COBRA is also smaller than the formulations in [105] and [114].
30. G. Cornuéjols, M.L. Fisher, and G.L. Nemhauser, Location of bank accounts to optimize float: An analytical study of exact and approximate algorithms, Manage Sci 23 (1977), 789-810. Presents a variety of solution techniques, including relaxations and heuristics. Also presents upper bounds for the worst case performance of the greedy interchange heuristic and the relaxations. The main result is that the relative error of the dual bound and the greedy heuristic is bounded above by 1/e.
31. E.S. Correa, M.T.A. Steiner, A.A. Freitas, and C. Carnieri, A genetic algorithm for solving a capacitated p-median problem, Numer Algorithms 35 (2004), 373-388. Develops a genetic algorithm for the capacitated p-median problem. The process assigns vertices to the nearest median that is not already full. A new genetic operator called heuristic hypermutation is introduced. This operator improves the fitness of a certain percentage of genes. Computational results of the algorithm with and without heuristic hypermutation are compared with the results of a tabu search heuristic.
32. T.G. Crainic, M. Gendreau, P. Hansen, and N. Mladenović, Parallel variable neighborhood search for the p-median, Les Cahiers du GERAD G-2003-4 (2003). Summarizes the methods in [55] and offers a new strategy entitled Cooperative Neighborhood VNS (CNVNS). This is a master-slave procedure where the master process initiates several VNS threads, each of which randomly chooses a neighborhood to explore. Threads report improved solutions to the master process in order to find an overall solution.
33. T.G. Crainic, M. Gendreau, P. Hansen, and N. Mladenović, Cooperative parallel variable neighborhood search for the p-median, J Heuristics 10 (2004), 293-314. Describes the Cooperative Neighborhood VNS (CNVNS) strategy found in [32].
34. M.S. Daskin, Network and discrete location: Models, algorithms, and applications, John Wiley & Sons, Inc., New York, 1995. Contains a chapter entitled "Median Problems," which describes in detail three types of heuristics: myopic (greedy), exchange (vertex substitution), and neighborhood search. A Lagrangian relaxation approach is also presented. A software implementation of these methods accompanies the book, and computational results of these approaches are given.
35. I.R. de Parias, Jr., A family of facets for the uncapacitated p-median polytope, Oper Res Lett 28 (2001), 161-167. Presents a nontrivial family of facet-defining inequalities for the convex hull of the feasible set of the p-median problem. The author develops a separation heuristic for this family of inequalities and then uses the inequalities as cuts in a branch-and-cut algorithm. For many problem instances, this method dramatically reduced the number of nodes. For small instances, however, it was inefficient.
36. P.J. Densham and G. Rushton, Designing and implementing strategies for solving large location-allocation problems with heuristic methods, Technical report 91-10, National Center for Geographic Information and Analysis, Buffalo, NY, 1991. Discusses how to implement vertex substitution [126], specifically detailing speedup strategies. These include minimizing the volume of data and access times to that data, and exploiting the spatial structure of the problem to reduce the number of vertices to check after each substitution. The authors also discuss using an allocation table as an alternative method of keeping track of the changes in the objective function as vertices are substituted.
37. P.J. Densham and G. Rushton, A more efficient heuristic for solving larger-median problems, Papers in Regional Sci 71 (1992), 307-329. Uses necessary conditions of optimal p-median solutions to develop a more efficient variant of vertex substitution [126]. This information is used to create an informed spatial search procedure that is more efficient and more effective than the original naive spatial search procedure. This procedure is implemented in a new method called Global/Regional Interchange Algorithm (GRIA).
38. P.J. Densham and G. Rushton, Strategies for solving large location-allocation problems by heuristic methods, Environ Plan A 24 (1992), 289-304. Provides a method for solving large location-allocation problems, including the p-median problem. The method is based on vertex substitution [126] and works by exploiting the spatial structure of location-allocation problems.
39. J.A. Dfaz and E. Fernández, Hybrid scatter search and path relinking for the capacitated p-median problem, Eur J Oper Res 169 (2006), 570-585. Gives three metaheuristic techniques for the capacitatedp-median variation. The first uses scatter search, the second involves path relinking, and the third combines both techniques. A GRASP (Greedy Randomized Adaptive Search Procedure) similar to the technique in [104] is used to find the initial reference set for each of these three heuristics. The authors present computational results, and claim that the combination of scatter search and path relinking produces the best results.
40. E. Domfnguez and J. Muñoz, Applying bio-inspired techniques to the p-median problem, IWANN 2005: Computational Intelligence Bioinspired Syst, 8th Int Workshop Artificial Neural Networks, Lecture Notes in Comput Science, Springer-Verlag, Berlin, 2005, pp. 67-74. Presents a genetic algorithm and a stochastic neural network. The genetic algorithm uses some specialized operations such as bit-flip mutation and uniform recombination. The stochastic neural network is based upon the recurrent neural network but is claimed to be more accurate here. The experimental results compare the genetic algorithm and both stochastic and recurrent neural networks to benchmarks. The authors note that on medium-scale instances, the neural networks outperform the genetic algorithm.
41. W. Domschke and A. Drexel, Location and layout planning: An international bibliography, Lecture Notes in Economics and Mathematical Systems 238, Springer-Verlag, Berlin, 1985. Lists over 1800 references pertaining to location theory.
42. O. du Merle and J.P. Vial, Proximal ACCPM, a cutting plane method for column generation and Lagrangian relaxation: Application to the p-median problem, Technical report 2002.23, HEC Genève, Université de Genève, 2002. Presents a variant of the analytic center cutting plane method (ACCPM) that incorporates two features from the Bundle method. A proximal term is added to the logarithmic barrier function and a step to reduce the number of columns in the localization set is implemented. Extensive computational results are presented.
43. J. Dvorett, Compatibility-based genetic algorithm: A new approach to the p-median problem, Technical report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1999. Presents a compatibility measure that helps genetic algorithms to make better guesses when selecting solutions for the reproduction and crossover stages. The compatibility measure considers candidate parents together rather than choosing the two parents independently. Computational results are presented, comparing this method with greedy heuristics [30, 78], vertex substitution [126], and the alternate heuristic [87]. The authors show that in some instances, the genetic algorithm outperforms other heuristics.
44. M. Efroymson and T. Ray, A branch-bound algorithm for plant location, Oper Res 14 (1966), 361-368. Provides an early integer programming formulation of the plant location problem that has been historically adapted to the p-median problem. This paper presents a branch-and-bound technique that is used to solve the relaxed linear program.
45. p with λ vertices in V no reduction in the objective value can be obtained. This is an extension of vertex substitution [126]. The calculation effort increases rapidly with λ and the authors warn against using algorithms with λ > 2. They also combine this method with the vertex addition method in [72]. This method iteratively calculates the 1,2,...,p-medians, applying the λ-optimal heuristic at each iteration. Extensive computational results are presented.
46. A.M. El-Shaieb, A new algorithm for locating sources among destinations, Manage Sci 20 (1973), 221-231. Presents two different ways of calculating lower bounds on total weighted distance to be used in branch-and-bound. Both methods begin with an allocation set {S : D} consisting of a set of sources S and a set of destinations D. These sets are empty at first, and at each iteration a location is added to either set, depending on which offers the least lower bound. The corresponding allocation set and its complement are added to the active set. This process continues until the number of sources equals p or the number of destinations equals (n - p). Once either of these occur, the algorithm calculates the total weighted distance and purges any allocation sets. The process continues until there is only one allocation set in the active set, which is the final solution.
47. D. Erlenkotter, A dual-based procedure for uncapacitated facility location, Oper Res 26 (1978), 992-1009. Presents a dual ascent algorithm commonly known as DUALOC for the UFLP. This method is often applied to thep-median problem. It balances the fixed costs of the median vertices with the variable costs of allocation when finding the optimal solution. To apply this algorithm to thep-median problem one must first find a fixed cost value that results in the desired number of p median vertices. Unfortunately, such a fixed cost may not exist, so solutions for all values of p cannot be guaranteed by this procedure.
48. V. Estivill-Castro and R. Torres-Veldzquez, Hybrid genetic algorithm for solving the p-median problem, SEAL '98: Selected Papers from Second Asia-Pacific Conference Simulated Evolution Learning, Springer-Verlag, Berlin, 1999, pp. 19-25. Combines genetic algorithms and vertex substitution [126]. This hybrid technique is useful for avoiding local optima that vertex substitution is prone to finding. The authors claim that this method outperforms both ordinary genetic algorithms and stand-alone vertex substitution.
49. J. Fathali and H.T. Kakhki, Solving the p-median problem with pos/neg weights by variable neighborhood search and some results for special cases, Eur J Oper Res 170 (2006), 440-462. Applies the Variable Neighborhood Search (VNS) technique in [65] to thep-median problem with vertex weights unrestricted in sign. This is different from the original formulation, where the weights are assumed to be positive. Extensive computational results are presented, comparing the VNS technique to greedy and vertex substitution heuristics, tabu search, and tabu search combined with vertex substitution.
50. ij ≥ 0, 1, j = 1,..., n. A heuristic to solve the dual of this LP relaxation is presented. This method is similar to the method in [47], but is specifically for the p-median problem. The optimal dual solution is used in a branch-and-bound algorithm.
51. R.D. Galvão, A graph theoretical bound for the p-median problem, Eur J Oper Res 6 (1981), 162-165. Finds lower bounds for a branch-and-bound algorithm. For an undirected network the length of the shortest spanning tree minus the shortest spanning tree's (p - 1) longest links is shown to be a lower bound on the solution. This bound is generalized for directed networks, where the author develops a stronger bound based upon the spanning arborescences of the network.
52. R.D. Galvão, A note on Garfinkel, Neebe, and Rao's LP decomposition for the p-median problem, Transportation Sci 15 (1981), 175-182. Presents extensive computational results of the LP decomposition method in [58]. Due to the method's degenerate nature, serious convergence problems commonly occurred. Convergence surprisingly occurred more often with randomly generated initial solutions rather than "good" initial solutions found with heuristics.
53. R.D. Galvão, Rejoinder to "A note on Galvão's 'A graph theoretical bound for the p-median problem,'" Eur J Oper Res 17 (1984), 128. The author acknowledges the error in [51] that was pointed out in [96], but emphasizes that the error does not affect his results.
54. R.D. Galvão and L.A. Raggi, A method for solving to optimality uncapacitated location problems, Ann Oper Res 18 (1989), 225-244. Presents a three-stage procedure for solving the p-median problem. The stages are (1) a primal-dual algorithm, (2) subgradient optimization to solve a Lagrangian dual, and (3) a branch-and-bound algorithm. The method is hierarchical, meaning that stages are only activated if the optimal solution was not found in the previous stage.
55. F. García-López, B. Melián-Batista, J.A. Moreno-Pérez, and J.M. Moreno-Vega, The parallel variable neighborhood search for the p-median problem, J Heuristics 8 (2002), 375-388. Offers three parallel computing strategies for speeding up the computation time of a Variable Neighborhood Search (VNS) metaheuristic for the p-median problem. VNS finds an initial local minimum and then systematically or randomly explores increasingly distant neighborhoods. If the procedure finds a better solution, it jumps to it and continues the search until a stopping criterion is met. The first strategy presented here involves parallelizing the local search phase of the algorithm. The second approach involves running an independent VNS procedure on each processor and using the best solution at the end. The third method involves a synchronous master-slave approach.
56. F. García-López, B. Melián-Batista, J.A. Moreno-Pérez, and J.M. Moreno-Vega, Parallelization of the scatter search for the p-median problem, Parallel Comput 29 (2003), 575-589. Describes a population-based metaheuristic technique known as scatter search. This method begins by creating a reference set from a population of solutions and generating subsets of this reference set that are good solutions over the reference set. These solutions are combined to form a new current solution. This solution is then run through a solution improvement procedure. A reference set updating procedure then considers the improved solutions for inclusion in the reference set. Stopping procedures determine when to generate a new reference set or a new population and when to terminate the algorithm.
57. M.R. Garey and D.S. Johnson, Computers and intractibility: A guide to the theory of NP-completeness, W.H. Freeman and Co., San Francisco, 1979. Shows that the p-median problem is NP-hard. If p is fixed then it is solvable in polynomial time. It is also solvable in polynomial time for arbitrary p if the network is a tree.
58. R.S. Garfinkel, A.W. Neebe, and M.R. Rao, An algorithm for the m-median plant location problem, Transportation Sci 8 (1974), 217-236. Models the p-median problem as an integer program and solves the relaxed LP with a decomposition technique. The paper gives a method for resolving noninteger solutions that combines group theoretic and dynamic programming techniques. Computational results are presented, comparing this technique with branch-and-bound techniques similar to those in [74].
59. F. Glover, Tabu search for the p-median problem, unpublished manuscript, 1990. Proposes an implementation of tabu search for the p-median problem and details its associated interchange, strategic oscillation, candidate list, intensification and diversification strategies.
60. M.F. Goodchild and V.T. Noronha, Location-allocation for small computers, Monograph No. 8, Technical report, University of Iowa, Department of Geography, 1983. Implements several shortest path algorithms and ALLOC [116] in BASIC for solving location-allocation problems. The implementation was tested with favorable results on the p-median problem.
61. S.L. Hakimi, Optimum locations of switching centers and the absolute centers and medians of a graph, Oper Res 12 (1964), 450-459. Shows that the absolute median of a graph C is always located at a vertex of a graph. Thus, to find the optimum location for a switching center in a communication network one must only search the vertices of the graph or network. The absolute median is equivalent to a 1-median.
62. S.L. Hakimi, Optimum distribution of switching centers in a communication network and some related graph theoretic problems, Oper Res 13 (1965), 462-475. Generalizes the results in [61] and shows that the optimum collection of p switching centers in a communication network is the p-median of the corresponding weighted graph. Hakimi proves that there is a set of p points, consisting entirely of vertices in a graph, that minimizes the total weighted cost. Hakimi calls this set the p-median of a graph and shows that to find it one may examine every p-element subset of the vertices of G and simply keep track of the least weighted distance after every evaluation. He then uses this method, known as direct enumeration, to find the 3-median of a graph with 10 vertices.
63. G.Y. Handler and P.B. Mirchandani, Location on networks: Theory and algorithms, MIT Press, Cambridge, MA, 1979. Provides an excellent overview of the computational methods for the p-median problem up to 1979. The solution methods are classified into five different categories: (1) enumeration, (2) graph theoretic, (3) heuristic, (4) primal-based LP methods, and (5) dual-based LP methods.
64. P. Hanjoul and D. Peeters, A comparison of two dual-based procedures for solving the p-median problem, Eur J Oper Res 20 (1985), 387-396. Presents and compares two dual-based p-median solution methods that rely on two different Lagrangian relaxations. Constraints (1) and (2), described in Section 2.3, are relaxed. Computational results indicate that these two procedures solve large-scale p-median problems successfully.
65. P. Hansen and N. Mladenović, Variable neighborhood search for the p-median, Location Sci 5 (1997), 207-226. Variable neighborhood search (VNS) is a metaheuristic that involves a systematic change of neighborhood within a local search algorithm. The process involves exploring increasingly distant neighborhoods to avoid local minima. The authors provide a parameter-free VNS heuristic for the p-median problem and compare it to tabu search and the greedy interchange heuristic.
66. P. Hansen, N. Mladenović, and D. Perez-Brito, Variable neighborhood decomposition search, J Heuristics 7 (2001), 335-350. Presents a variant of Variable Neighborhood Search (VNS) designed to enhance the efficiency of VNS on larger problems. The only difference between VNS and the new method, called Variable Neighborhood Decomposition Search (VNDS), is that during the local search phase, VNDS solves a subproblem instead of applying the local search to the whole solution space. This method is applied to the p-median problem and computational results are presented comparing VNDS with the fast interchange method in [131] and a Reduced VNS method.
67. M. Horn, Analysis and computational schemes forp-median heuristics, Environ Plan A 28 (1996), 1699-1708. Provides a mathematical analysis of a computational scheme for the vertex substitution heuristic [126] and GRIA [37]. The author decided to use concise settheory notation rather than the binary-array notation used in [37]. It is shown that vertex substitution and GRIA do not generally find solutions with equivalent local optimality.
68. C.M. Hosage and M.F. Goodchild, Discrete space locationallocation solutions from genetic algorithms, Ann Oper Res 6 (1986), 35-46. Provides the first application of genetic algorithms to the p-median problem. The authors note strengths and weaknesses of this implementation, showing that it is likely to be trapped in a local optimum when solving location problems.
69. M. Hribar and M.S. Daskin, A dynamic programming heuristic for the p-median problem, Eur J Oper Res 101 (1997), 499-508. Provides a heuristic that is a cross between a greedy heuristic [78] and a dynamic programming algorithm. This polynomial time method finds several solutions and determines how often particular points are used in solutions, thus finding points that have a high or low likelihood of being in the optimal solution. Computational results are presented, comparing the solutions obtained with this method to known optimal solutions for several test problems.
70. K. Jain, M. Mahdian, and A. Saberi, A new greedy approach for facility location problems, STOC '02: Proc Thirty-Fourth Ann ACM Symp Theory Comput, New York, 2002, pp. 731-740. Presents a lower bound on the approximability of the metric p-median problem, showing that it may not be approximated with a factor strictly smaller than 1 + 2/ε.
71. ij, or the metric distance.
72. P. Järvinen, J. Rajala, and H. Sinervo, A branch-and-bound algorithm for seeking the p-median, Oper Res 20 (1972), 173-178. Applies a branch-and-bound technique to finding the p-median by attempting to find the (n - p) vertices that are not in the p-median. It does so by starting from an availiable (t - 1)-median solution and adding to it the vertex that provides the maximum reduction in the objective value as (t - 1) → t. This process begins with (t - 1) = 1 and ends when t = (n - p). The insight is that the removed vertices would not be a good choice for the p-median because their removal results in a reduction in the total weighted distance. Extensive computational results are presented.
73. 2) algorithm for the p-median problem on a tree where p is arbitrary.
74. ij are integers, then Z* is the final solution to the problem without relaxation. If fractional solutions are found, branch-and-bound is used to eliminate them.
75. B.M. Khumawala, A.W. Neebe, and D.G. Dannenbring, A note on El-Shaieb's new algorithm for locating sources among destinations, Manage Sci 21 (1974), 230-233. Compares the solution method in [46] with the methods in [58], [74], and [126].
76. 6 n). This is nearly linear for any fixed ε > 0 and d.
77. M.R. Korupolu, C.G. Plaxton, and R. Rajaraman, Analysis of a local search heuristic for facility location problems, J Algorithms 37 (2000), 146-188. Presents a summary of the work done in the field of approximation algorithms for the p-median problem. This paper deals only with the metric p-median problem. The authors show that the local search technique in [78] yields a polynomial time algorithm that, for any ε > 0, computes a solution using at most (3 + (5/ε))p medians with cost at most (1 + ε) times the cost of an optimal solution with at most p facilities.
78. A.A. Kuehn and M.J. Hamburger, A heuristic program for locating warehouses, Manage Sci 9 (1963), 643-666. Introduces a greedy heuristic for the warehouse location problem that has historically been applied to the p-median problem. Let M be the set of potential warehouse locations and let N be the number of locations to evaluate at each iteration. The greedy heuristic initially chooses N locations that maximize the cost savings of replacing these N locations with warehouses. It then considers each of these locations individually and calculates the total distribution cost. Any location that does not reduce the total cost is eliminated from further consideration. The location that gives the least cost is assigned a warehouse, and any remaining locations go back to the list of possible locations to test. This process is repeated until all elements of the original list of potential warehouses have either been eliminated or assigned as a warehouse.
79. M. Labbé, D. Peeters, and J. Thisse, "Location on networks", Handbooks in operations research and management science, 8: Network routing, M.O. Ball, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser (Editors), North-Holland, Amsterdam, 1995, pp. 551-624. Provides an excellent theoretical survey of network location problems. The authors discuss the formulation and complexity of thep-median problem as well as several exact and heuristic solution methods.
80. T.V. Levanova and M.A. Loresh, Algorithms of ant system and simulated annealing for thep-median problem, Automat Remote Control 65 (2004), 431-438. Explores the implementation of an ant system and a simulated annealing algorithm. Ant system algorithms were suggested by the ability of ants to find the shortest path from an ant hill to food by using pheromones. The theory is that over time, the shortest path will have the greatest amount of pheromone, and will therefore be the most probable path. The authors use this insight and a simulated annealing algorithm to solve thep-median problem and present computational results.
81. A. Lim and Z. Xu, A fixed-length subset genetic algorithm for thep-median problem, Lecture Notes in Computer Science 2724 (2003), 1596-1597. The fixed-length subset genetic algorithm represents candidate solutions by a fixed-length subset. Actual computational results are not presented, but the authors claim that their method outperforms the traditional genetic algorithm and is able to find solutions very close to optimal for most problems.
82. J.H. Lin and J.S. Vitter, Approximation algorithms for geometric median problems, Informa Process Lett 44 (1992), 245-249. Deals only with the metric p-median problem. The authors present a polynomial time algorithm that, for any ε > 0, finds a solution of no more than 2(1 + ε) times the optimal cost and of at most (1 + (1/ε))p median vertices.
83. L.A.N. Lorena and J.C. Furtado, Constructive genetic algorithm for clustering problems, Evolut Comput 9 (2001), 309-328. The constructive genetic algorithm presented here differs from the traditional genetic algorithm in that it uses a dynamic population. Two separate fitness functions are described and the clustering problems are formulated as bi-objective optimization problems. Computational results are presented.
84. L.A.N. Lorena and E.L.F. Senne, Local search heuristics for capacitated p-median problems, Networks Spatial Econo 3 (2003), 409-419. Combines the Lagrangian/surrogate relaxation techniques in [119] with some local search heuristic techniques to produce a new method for solving the capacitated p-median problem. The heuristic techniques are used to improve solutions that are made feasible by the Lagrangian/surrogate process and involve swapping medians within clusters and reallocating vertices. This is repeated until no further improvement is made.
85. L.A.N. Lorena and E.L.F. Senne, A column generation approach to capacitated p-median problems, Comput Oper Res 31(2004), 863-876. Applies the column generation and Lagrangian/surrogate techniques in [120] to the capacitated p-median problem.
86. V. Maniezzo, A. Mingozzi, and R. Baldacci, A bionomic approach to the capacitated p-median problem, J Heuristics 4 (1998), 263-280. Presents a metaheuristic technique similar to a genetic algorithm for the capacitated p-median problem. The technique is known as a bionomic algorithm and differs from genetic algorithms in how the parent set is obtained. Extensive computational results are presented.
87. RE. Maranzana, On the location of supply points to minimize transport costs, Oper Res Q 15 (1964), 261-270. Presents a heuristic for finding the p-median that, after each iteration, yields a collection of p vertices that is guaranteed to either reduce or leave unchanged the total weighted distance. The algorithm begins with an arbitrary p vertices of the network and partitions the remaining vertices into corresponding nearest neighborhood cells. The algorithm then determines a "center of gravity" for each partition cell. The collection of these points is a solution to the p-median problem. The author shows that with respect to each iteration, the total weighted distance D is monotonically nonincreasing, but the algorithm does not always converge to an optimum.
88. R.E. Marsten, An algorithm for finding almost all the medians of a network, Technical report 23, Center for Math Studies in Economics and Management Science, Northwestern University, 1972. Shows that for a network of n vertices every p-median (1 ≤ p ≤ n) is an extreme point of a polyhedron. An algorithm that tours these extreme points is presented. Some extreme points correspond top-median solutions with a fractional value of p. Furthermore, the tour may not hit a p-median for every value of p. Aside from these exceptions the algorithm is shown to create a complete set of medians.
89. E.D. Merino and J.M. Pérez, An efficient neural network algorithm for the p-median problem, IBERAMIA 2002: Proceedings 8th Ibero-American Conference AI, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 2002, pp. 460-469. Presents an alternate formulation of the p-median problem and then applies a Hopfield neural network model to solve it. The problem formulation involves two types of neurons: one for location and the other for allocation. The constrained formulation is turned into an unconstrained problem by introducing a penalty function that penalizes the objective function if constraints are violated. Computational results are presented comparing this method with vertex substitution [126].
90. E.D. Merino, J.M. Pérez, and J.J. Aragones. Neural network algorithms for the p-median problem, Proc 2003 ur Symp Artificial Neural Networks (ESANN 2003), D-Facto, Brussels, 2003, pp. 385-391. Offers three different variations of a neural network algorithm for the p-median problem. The objective function is modelled as an energy function, which is guaranteed to decrease or remain unchanged as the system changes according to a given dynamical rule. The different variations, iterative, agglomerative, and stepwise, are tested against vertex substitution [126] and the results are presented.
91. 2 n, the complexity is shown to reduce to O(np).
92. P.B. Mirchandani, "The p-median problem and generalizations", Discrete location theory, P.B Mirchandani and R. Francis (Editors), John Wiley & Sons, Inc., New York, 1990, pp. 55-117. Describes the classical p-median problem and discusses variations such as the p-median problem on oriented networks, probabilistic costs and demands, and multidimensional networks. The author then gives a survey of solution methods, classifying them into the same five categories as [63].
93. P.B. Mirchandani, A. Oudjit, and R.T. Wong, "Multidimensional" extensions and a nested dual approach for the m-median problem, Eur J Oper Res 21 (1985), 121-137. Describes a "nested dual" approach for solving the p-median problem that uses the method in [47] as a subroutine. The problem is first dualized with respect to Constraint (2) (see Section 2.3). The Lagrangian dual is then solved by a simplex method, where the method in [47] is used to solve the Lagrangian subproblems.
94. N. Mladenović, J. Brimberg, P. Hansen, and J.A. Moreno-Pérez, The p-median problem: A survey of metaheuristic approaches, Les Cahiers du GERAD G-2005-39 (2005). Provides an overview of some of the most common heuristic and metaheuristic approaches applied to the p-median problem.
95. J.A. Moreno-Pérez, J.L. Roda Garcfa, and J.M. Moreno-Vega, A parallel genetic algorithm for the discretep-median problem, Stud Local Anal 7 (1994), 131-141. Implements a variant of the genetic algorithm that does not involve the traditional binary string solution representation (chromosome), but instead represents solutions by a collection of indices of the demand points in V. This algorithm is known as an evolutive algorithm, and it is implemented in parallel, using software to create a parallel virtual machine on a local network. The solution is split into several colonies, each of which is sent to a different processor. This method was tested on TSP data.
96. L. Morgenstern, A note on Galvão's "A graph theoretical bound for thep-median problem," Eur J Oper Res 12 (1983), 404-405. Points out that a step of a bound improvement algorithm in [51] is flawed.
97. ij = 1, for j = 1,...,n, is relaxed (i.e., absorbed into the Lagrangian). The algorithm begins with a "good" initial value for the dual variable and moves in a direction where the subgradient is not zero. The algorithm terminates when all the components of the subgradient are zero or when there is no duality gap. Computational results comparing this with the methods in [46], [74], [105], and [126] are presented.
98. A.W. Neebe and M.R. Rao, A subgradient approach to the m-median problem, Technical report 75-12, University of North Carolina, Chapel Hill, 1975. Presents a subgradient method for solving the dual of the relaxed LP to overcome degeneracy in decomposition techniques. This procedure often terminates with the maximum dual objective value, and in some cases it leads to integer values for all of the primal variables. If not, the bounds may be used in a branch-and-bound algorithm.
99. H. Pirkul, R. Gupta, and E. Rolland, VisOpt: A visual interactive optimization tool for p-median problems, Dec Support Syst 26 (1999), 209-223. Introduces a visual software tool for finding good solutions by attempting to
100. N.D. Pizzolato, A heuristic for large-size p-median location problems with application to school location, Ann Oper Res 50 (1994), 473-485. Introduces a heuristic that begins with an initial collection of p trees generated by the technique in [87] and then reshapes them iteratively through a root interchange method. These steps are fast and provide a near-optimal solution. The author then applies some slower techniques for improving the final solution. This method is compared with vertex substitution [126].
101. S. Rahman and D.K. Smith, A comparison of two heuristic methods for the p-median problem with and without maximum distance constraints, Int J Oper Product Manage 11 (1991), 76-84. Compares vertex substitution [126] with a heuristic originally developed for a service facility location problem, found in [3]. The authors call it the Ardalan heuristic. Unlike vertex substitution, once a vertex is chosen as a median, it remains in the solution set until termination. Computational results are presented showing that vertex substitution consistently outperformed the Ardalan heuristic on test instances.
102. M.G.C. Resende and R.F. Werneck, On the implementation of a swap-based local search procedure for the p-median problem, Proc Fifth Workshop Algorithm Eng Experiments (ALENEX), SIAM, Philadelphia, 2003, pp. 119-127. Develops a more efficient variant of vertex substitution [126] known as fast interchange. Several techniques to hasten the algorithm are developed. These include storing partial results in a matrix to speed up later steps, compressing this matrix, and preprocessing techniques. They report obtaining speedups of up to three orders of magnitude over the original fast interchange method.
103. M.G.C. Resende and R.F. Werneck, A fast swap-based local search procedure for location problems, Technical report TD-5R3KBH, AT&T Labs Research, 2004. Provides an implementation of vertex substitution [126] that builds on the implementation in [131]. This method uses gain, loss, extra, and netloss functions to determine the value of different substitutions. Values for these functions are stored in data structures so that the best swap may easily be found.
104. M.G.C. Resende and R.F. Werneck, A hybrid heuristic for the p-median problem, J Heuristics 10 (2004), 59-88. Introduces a randomized multistart iterative metaheuristic known as GRASP (Greedy Randomized Adaptive Search Procedure). Each iteration of this process applies a greedy randomized algorithm followed by a local search procedure. A pool of some of the best solutions of previous iterations is stored, and after each iteration, the new candidate solution is combined with one stored solution in a process called path-relinking. Once the algorithm terminates, the stored solutions are combined with each other.
105. C. ReVelle and R. Swain, Central facilities location, Geographi Anal 2 (1970), 30-42. Provides the first linear programming formulation of the p-median problem. Constraint (4), described in Section 2.3, is relaxed. The authors recommend using a branch-and-bound technique when dealing with fractional solutions. This formulation was tested on several problem instances and no fractional solutions occurred.
106. G. Righini, A double annealing algorithm for discrete location/allocation problems, Eur J Oper Res 86 (1995), 452-468. Presents a double annealing algorithm and tests it on instances of the p-median problem. The algorithm is a variant of mean-field annealing, which is a deterministic version of simulated annealing. This method splits the annealing process into two synchronized parallel processes. The author shows that if a deannealing process is used, allowing the annealing temperature to increase instead of just decrease, then the algorithm experimentally improves.
107. E. Rolland, D.A. Schilling, and J.R. Current, An efficient tabu search procedure for thep-median problem, Eur J Oper Res 96 (1996), 329-342. Tabu search is a metaheuristic designed to be used in conjunction with heuristics that move vertices between sets, such as vertex substitution [126]. The tabu search method described here involves tabu restrictions, aspiration criteria, diversification, and strategic oscillation. The restrictions prevent the search from moving back to previous solutions. Aspiration criteria allow the search to move a vertex even if it has been restricted. Diversification is used to escape from a local minimum by deterring the search from performing the same moves too often. Strategic oscillation allows the search to intentionally pass through infeasible solutions in order to avoid local optima. Computational results are presented and this method is compared to the method in [37].
108. K.E. Rosing, An empirical investigation of the effectiveness of a vertex substitution heuristic, Environ Plan B 24 (1997), 59-67. Presents a comparison of optimal solutions to solutions found with the vertex substitution implementation in [116] on 90 test problems. The values of n and p were varied systematically, and the results showed a degradation of solution quality as either n or p increased.
109. K.E. Rosing, E.L. Hillsman, and H. Rosing-Vogelaar, A note comparing optimal and heuristic solutions to the p-median problem, Geograp Anal 11 (1979), 86-89. Compares the optimal solution of six test problems to solutions found with the vertex substitution implementation in [116] and solutions found with the alternate heuristic [87]. The vertex substitution heuristic found the optimal solution more frequently than the alternate heuristic.
110. K.E. Rosing, E.L. Hillsman, and H. Rosing-Vogelaar, The robustness of two common heuristics for the p-median problem, Environ Plan A 11 (1979), 373-380. Compares the solutions found using the alternate heuristic [87] and vertex substitution [126] with known optimal solutions for six test problems. The authors found that the quality of solutions generated by the alternate heuristic decreases rapidly as p grows. Vertex substitution was more stable.
111. K.E. Rosing and M.J. Hodgson, Heuristic concentration for the p-median: An example demonstrating how and why it works, Comput Oper Res 29 (2002), 1317-1330. Uses combinatorial and map analysis to show how heuristic concentration [112] can be more effective than vertex substitution [126]. The authors show how vertex substitution may get stuck in traps of particular groups of vertices that heuristic concentration avoids. Good vertex substitution solutions only have one such trap, so a set of several of these solutions is likely to identify' all of the vertices required to find the optimum. If all of the optimal vertices are captured in the concentration set, then the second stage of heuristic concentration is able to find the optimal solution.
112. K.E. Rosing and C.S. ReVelle, Heuristic concentration: Two stage solution construction, Eur J Oper Res 97 (1997), 75-86. The method presented here is a two stage heuristic. In the first stage, the method analyzes several solutions found with vertex substitution [126] and creates a concentration set that contains vertices that have a high probability of being the facilities in the optimal solution. The second stage involves using an exact algorithm to solve a subproblem on this set.
113. K.E. Rosing, C.S. ReVelle, E. Rolland, D.A. Schilling, and J.R. Current, Heuristic concentration and tabu search: A head to head comparison, Eur J Oper Res 104 (1998), 93-99. Presents computational results comparing tabu search [107] with heuristic concentration [112]. The authors report that heuristic concentration found the superior solution in about 95% of the test instances. When the optimal solution was previously known, it found the optimal solution in about 80% of the cases.
114. K.E. Rosing, C.S. ReVelle, and H. Rosing-Vogelaar, The p-median and its linear programming relaxation: An approach to large problems, J Oper Res Soc 30 (1979), 815-823. Summarizes the plant location IP formulations in [11] and [44], noting the superiority of the former in terms of terminating with few noninteger solutions. It also cites [105] as the standard formulation of the p-median problem. Techniques that use these formulations to solve larger problems are developed. These involve removing columns or rows from the fully specified problem.
115. K.E. Rosing, C.S. ReVelle, and D.A. Schilling, A gamma heuristic for the p-median problem, Eur J Oper Res 117 (1999), 522-532. Presents a heuristic that is a variant of heuristic concentration [112]. The first stage is the same as heuristic concentration, but the second stage applies a 2-opt procedure that assures no exchange of any two median vertices for any two nonmedian vertices results in further reduction of the objective function. This is followed by a 1-opt procedure. The entire process is a metaheuristic technique called the gamma heuristic.
116. G. Rushton and J.A. Kohler, "ALLOC: Heuristic solutions to multifacility location problems on a graph," Computer programs for location-allocation problems, G. Rushton, M. Goodchild, and L. Ostresh (Editors), Monograph No. 6, University of Iowa, Department of Geography, 1973, pp. 163-187. Presents a FORTRAN implementation of two common heuristics for solving the p-median problem. The alternate heuristic [87] and vertex substitution [126] are implemented. The authors note that in 75 test runs, vertex substitution almost invariably outperformed the alternate heuristic.
117. S. Salhi, A perturbation heuristic for a class of location problems, J Oper Res Soc 48 (1997), 1233-1240. Presents an algorithm that relaxes the constraint on the cardinality of thep-median solution with the goal of either adding or subtracting vertices to obtain a feasible solution. Letting q ≤ p, where q is a fixed integer, this method solves two relaxed problems to obtain the (p + q-median and the (p - q)-median. When a solution for the (p - q)-median is found, the cardinality of the solution is small enough to nearly guarantee that these vertices will remain in the final solution. Similarly, by dropping some of the vertices in the (p + q)-median, a better feasible solution is likely to be found. This process is repeated several times, and has a filtering effect where the most effective vertices tend to remain in the best configuration. After a fixed amount of time or after no better solution may be found, a diversification strategy is used to guide the search to other regions that may not be reached otherwise.
118. S. Salhi, Defining tabu list size and aspiration criterion within tabu search methods, Comput Oper Res 29 (2002), 67-86. Develops a functional representation of the tabu list size, allowing for a dynamic tabu list size. The author develops softer aspiration criteria that take into account (1) the tabu status of the attribute for that solution, (2) how much a solution differs from the best found so far, and (3) the change in the objective function. Computational results are presented.
119. E.L.F. Senne and L.A.N. Lorena, "Lagrangean/surrogate heuristics for p-median problems," Computing tools for modeling, optimization and simulation: Interfaces in computer science and operations research, M. Laguna and J. Gonzalez-Velarde (Editors), Kluwer Academic Publishers, Boston, 2000, pp. 115-130. Relaxes Constraint (1) (see Section 2.3) using a surrogate relaxation. This relaxation is not easily solved, so a Lagrangian relaxation is used on the surrogate formulation of the problem. Heuristic techniques are used to find a range of Lagrangian/surrogate multiplier values that improve the bounds of the usual Lagrangian relaxation technique. This approach was able to generate approximate solutions at least as good as the traditional Lagrangian relaxation technique while reducing computational effort for larger problems.
120. E.L.F. Senne and L.A.N. Lorena, Stabilizing column generation using Lagrangean/surrogate relaxation: An application to p-median location problems, Unpublished manuscript, 2001. Describes relationships between the Lagrangian/surrogate relaxation technique in [119] and column generation. This paper applies a column generation approach to the p-median problem that uses Lagrangian/surrogate relaxation as an acceleration process, generating new productive sets of columns at each iteration. Computational results show some improvement over the traditional column generation techniques.
121. E.L.F. Senne and L.A.N. Lorena, Complementary approaches for a clustering problem, XI CLAIO: Latin-Iberian American Congress Oper Res, University of Concepción, Chile, 2002, Proceedings on CD-ROM. Combines the Lagrangian/surrogate relaxation techniques in [119] with subgradient optimization and column generation to create two new heuristics. Computation results are presented, and the authors note that the heuristic using column generation is best when used on large-scale instances. The subgradient heuristic performed better on small-scale instances.
122. E.L.F. Senne, L.A.N. Lorena, and M.A. Pereira, A branch-and-price approach top-median location problems, Comput Oper Res 32 (2005), 1655-1664. Combines the traditional column generation approach with Lagrangian/surrogate relaxation. The method is a tree search algorithm employing column generation at each search node. The authors claim that the Lagrangian/surrogate multiplier method modifies the reduced cost criterion so that more productive columns are selected at each search node than in the traditional column generation method. The authors also claim that the algorithm is faster than the traditional approach.
123. E.D. Taillard, Heuristic methods for large centroid clustering problems, Technical report 96-96, Istituto Dalle Molle Di Studi Sull Intelligenza Artificiale, Switzerland, 1996. Applies three new clustering methods to solving the p-median problem-candidate list search (CLS), local optimization (LOPT), and decomposition/recombination (DEC). CLS begins with the alternate heuristic [87], obtaining a locally optimal solution. This solution is perturbed by eliminating a vertex and adding another, similar to vertex substitution [126]. The new solution is chosen only if it is better than the initial one. LOPT selects a median and some nearby medians and generates and solves a corresponding subproblem. DEC uses LOPT to obtain a good solution for the overall problem.
124. 2).
125. B.C. Tansel, R.L. Francis, and T.J. Lowe, Location on networks: A survey. I. The p-center and p-median problems, Manage Sci 29 (1983), 482-497. Contains an extensive theoretical analysis of the p-median problem as well as a small survey of solution methods.
126. i. If this location exists, then the vertices are switched, and the process is repeated on the new solution. When all vertices have been checked, the algorithm terminates with a local minimum solution.
127. M. Thorup, Quick it-median, it-center, and facility location for sparse graphs, ICALP '01: Proceedings 28th Int Colloq Automata, Languages Program, Springer-Verlag, Berlin, 2001, pp. 249-260. Presents a 12 + o(1) constant factor approximation algorithm for the p-median problem.
128. S. Voß, A reverse elimination approach for the p-median problem, Stud Local Anal 8 (1996), 49-58. Applies a tabu search procedure known as reverse elimination to the p-median problem. This procedure implements a necessary and sufficient condition so that known solutions are not revisited. This involves storing the entire search history in a running list. When used in conjunction with a diversification strategy, this was found to yield favorable results. Computational results are presented, comparing this method with the greedy interchange method in [131].
129. R.A. Whitaker, A tight bound drop exchange algorithm for solving the p-median problem, Environ Plan A 13 (1981), 669-680. Details a drop algorithm that starts with all n vertices in the solution set and an objective value of 0. On each iteration, k vertices are removed from the solution set and (k - 1) vertices are brought back into the set, yielding a net loss of one vertex per iteration. On each iteration, the procedure attempts to minimize the amount that the objective value increases. The algorithm terminates when p vertices remain in the solution set. Computational results are presented comparing this method to a greedy interchange method.
130. R.A. Whitaker, Some interchange algorithms for median location problems, Environ Plan B 9 (1982), 119-129. A variation of vertex substitution [126] that involves exchanging multiple vertices at once is presented. Several ways of producing initial solutions for this method are outlined, including greedy heuristics and drop algorithms (similar to the one presented in [129]). Extensive computational results are presented.
131. R.A. Whitaker, A fast algorithm for the greedy interchange of large-scale clustering and median location problems, INFOR 21 (1983), 95-108. Presents a method that initializes vertex substitution [126] with the solution obtained with a fast greedy algorithm based upon the methods in [30] and [78]. This method found solutions to test problems more than an order of magnitude faster than normal vertex substitution initialized with a normal greedy technique.
132. N.E. Young, Greedy approximation algorithms for k-medians by randomized rounding, Technical report PCS-TR99-344, Department of Computer Science, Dartmouth College, Hanover, NH, 1999. Applies randomized rounding and other probabilistic methods to understanding the operation of greedy approximation algorithms. An approximation algorithm is presented that, for any ε > 0, finds a solution using at most ln(n + (n/ε))p medians and with objective value no more than (1 + ε) times the optimal solution. This algorithm requires ln(n + (n/ε))p linear time iterations.